What an Accelerometer is

An accelerometer simply measures acceleration, either due to motion or due to gravity.

Acceleration is measured in m/s2.

The acceleration at near the surface of the earth is around 9.8m/s2 (or 32ft/s2), which can be an unwieldy number, so 9.8m/s2 is often labeled 1G.

This means that if you drop an object from some height (assuming a vacuum), after 1 second it will be falling at 9.8m/s, a speed of 19.6m/s after 2 seconds, and so on.

Formulae

Displacement after movement at an initial speed for a given time

s=u * t + 1/2 a * t2

initial speed * time + 1/2 * acceleration * time squared

Speed after Acceleration from rest

v=a * t

Speed = Acceleration * time

What one three-axis accelerometer can do

If the orientation of the phone is held constant - for example, clamped into a car holder, then you can make pretty good guesses about the correlation of aceleration and position.

In normal driving, the ground track of the car is some function of the position of the steering wheel. Any lateral accelerations against the current known course mean that the steering wheel has been applied, and the car is turning at a rate which can be computed by calculating it with the cars known speed.

Accelerations in line with the course mean that the car is slowing up or down, and any vertical accelerations mean that the car is going up or down a hill.

Errors build up, but it's not bad for short periods.

1cm/s^2 of error, with a car going in a straight line at 20m/s, takes a bit over a minute to move from a true direction of northeast - say, to a false direction of northnortheast. (the position is off after this period by a hundred metres or two)

When hand-held, and stationary, the phone has quite accurate (better than a degree) knowledge of where 'down' is.

Tilt in two directions can easily be calculated

Roll about the 'down' axis cannot.

What two three axis accelerometers can do

What can be done with absolutely perfect devices?

Imagine two three axis accellerometers rigidly placed on each end of an arrow.

How much information do you get out of these?

A few moments thought should reveal that the orientation of the sensors does not matter. You can resolve the three signals into one direction and size.
This does not change however the devices are oriented - there is no benefit in for example skewing one 45 degrees.

In Space!

Considering the case in deep-space - the maths are simpler.

(not a common use-case, but simpler to analyse)
Holding the orientation steady, you can do perfect inertial navigation, and determine exactly where you are at all times. (in a flat space-time, but meh)

What happens when we vary the orientation?

You can subtract any common accelleration that's measured by both sensors - this does not change the orientation.

Remembering that we can skew the sensors against each other, and this has no effect, let's specify that they are oriented with X and Y lined up, and Z pointing in the direction of the arrow.

This reveals a problem.

Spin the arrow on its axis, and none of the accelererometers measure anything at all - they do not move, so they do not accelerate.

(you can't get round this by moving them off-axis, as you can draw an imaginary arrow between the two accelerometers which has the same problem)

Spin the arrow around its centre (it must spin around its centre logically if you've subtracted the overall acceleration) you can pick up pitch and yaw.

With Gravity

Now, what if we add gravity in?

With perfect accelerometers again, with Z axes pointing to the arrow tip.

As long as the Z axes does not point in the same direction as gravity + current acceleration, then you can determine roll, pitch, yaw, and XYZ acceleration.

If the Z axes do point to the acceleration vector, then you lose track of roll.

In theory - with perfect accelerometers, this does not matter because you can never line it up perfectly.

In practice, with real ones, it gets more complex.

Roll signal/noise will drop as the acceleration vector closes on the Z axes, and be useless once it gets within the noise.

Numbers

I'm assuming specs similar to the ADXL330 - simplified a little.

Assumptions:

The Neo is a rigid object, and the accelerometers are rigidly fixed to it.

For interactive use.
High-pass filtering the accelerometer with a bandwidth of 10Hz - you can't filter it much more than that or you lose important 'wobbles', because you need to integrate them to come up with a position - leads to a noise floor of 300uG/sqrt(Hz) *sqrt(10Hz) = 1mG. (RMS (No, not that RMS))

Neglecting roll for the moment.

1mG is an acceleration of 1cm/s^2.

If the accelerometers are spaced 10cm apart, then the radius between each and the center is 5cm, meaning the circumference of the circle is 30cm.
Integrating over 1s, noise is around 3cm/s^2.

After 1s, if you happen to hit an average noise peak in each accelerometer at the opposite point - something that'll happen once every 5-10 seconds or so, (absolute peaks are much worse) what happens to the pointing?

Well - the velocity reads out as 6cm/s^2 wrong, which means that the position is now out by 3cm, or 10 degrees.

Results

What does this mean though?
Because the differential accelerations are so small, and comparable to the noise floor for gentle turning and twisting, there are basically two ways that they can be useful.

When more or less stationary:

Picking up turning about the gravity axis - if you are looking at the phone with it in front of you at chest level, with accelerometers at top and bottom for example, this will let you pick up sharp rotations of the phone around the vertical axis - keeping the angle the screen keeps with the floor constant.

Roll signal-noise drops as the phone long axis approaches vertical, becoming useless well before it is vertical.

Movements have to be sharp and fairly large.

It will probably not be good enough for keeping a map aligned when you turn in place.

It may be adequate for some games, but again, the movements have to be sharp, it will false signal as you decrease the wanted signal.

When in motion:

It may be adequate to improve positioning of the GPS, when sharply going round corners, or doing high-G aerobatics.

It can fill in _short_ - 3-5s gaps in GPS coverage.

In the best case - phone on its back, accelleration vector down, and turning in a vehicle, you may be able to tell sharp turns.

This is only useful if the phone is handheld, or not fixed in position to the vehicle, or if the vehicle does not go where it points.

And this is completely neglecting the temperature sensitivity.

What it can't do.

Inertial navigation over more than several seconds.

An example of an accelerometer that might be used is the one in the Wiimote, as documented in this page .
This page gives the sensitivity of the accelerometer as around 300ug/sqrt(Hz).

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What an Accelerometer is

An accelerometer simply measures acceleration, either due to motion or due to gravity.

Acceleration is measured in m/s2.

The acceleration at near the surface of the earth is around 9.8m/s2 (or 32ft/s2), which can be an unwieldy number, so 9.8m/s2 is often labeled 1G.

This means that if you drop an object from some height (assuming a vacuum), after 1 second it will be falling at 9.8m/s, a speed of 19.6m/s after 2 seconds, and so on.

Formulae

Displacement after movement at an initial speed for a given time

s=u * t + 1/2 a * t2

initial speed * time + 1/2 * acceleration * time squared

Speed after Acceleration from rest

v=a * t

Speed = Acceleration * time

What one three-axis accelerometer can do

If the orientation of the phone is held constant - for example, clamped into a car holder, then you can make pretty good guesses about the correlation of aceleration and position.

In normal driving, the ground track of the car is some function of the position of the steering wheel. Any lateral accelerations against the current known course mean that the steering wheel has been applied, and the car is turning at a rate which can be computed by calculating it with the cars known speed.

Accelerations in line with the course mean that the car is slowing up or down, and any vertical accelerations mean that the car is going up or down a hill.

Errors build up, but it's not bad for short periods.

1cm/s^2 of error, with a car going in a straight line at 20m/s, takes a bit over a minute to move from a true direction of northeast - say, to a false direction of northnortheast. (the position is off after this period by a hundred metres or two)

When hand-held, and stationary, the phone has quite accurate (better than a degree) knowledge of where 'down' is.

Tilt in two directions can easily be calculated

Roll about the 'down' axis cannot.

What two three axis accelerometers can do

What can be done with absolutely perfect devices?

Imagine two three axis accellerometers rigidly placed on each end of an arrow.

How much information do you get out of these?

A few moments thought should reveal that the orientation of the sensors does not matter. You can resolve the three signals into one direction and size.
This does not change however the devices are oriented - there is no benefit in for example skewing one 45 degrees.

In Space!

Considering the case in deep-space - the maths are simpler.

(not a common use-case, but simpler to analyse)
Holding the orientation steady, you can do perfect inertial navigation, and determine exactly where you are at all times. (in a flat space-time, but meh)

What happens when we vary the orientation?

You can subtract any common accelleration that's measured by both sensors - this does not change the orientation.

Remembering that we can skew the sensors against each other, and this has no effect, let's specify that they are oriented with X and Y lined up, and Z pointing in the direction of the arrow.

This reveals a problem.

Spin the arrow on its axis, and none of the accelererometers measure anything at all - they do not move, so they do not accelerate.

(you can't get round this by moving them off-axis, as you can draw an imaginary arrow between the two accelerometers which has the same problem)

Spin the arrow around its centre (it must spin around its centre logically if you've subtracted the overall acceleration) you can pick up pitch and yaw.

With Gravity

Now, what if we add gravity in?

With perfect accelerometers again, with Z axes pointing to the arrow tip.

As long as the Z axes does not point in the same direction as gravity + current acceleration, then you can determine roll, pitch, yaw, and XYZ acceleration.

If the Z axes do point to the acceleration vector, then you lose track of roll.

In theory - with perfect accelerometers, this does not matter because you can never line it up perfectly.

In practice, with real ones, it gets more complex.

Roll signal/noise will drop as the acceleration vector closes on the Z axes, and be useless once it gets within the noise.

Numbers

I'm assuming specs similar to the ADXL330 - simplified a little.

Assumptions:

The Neo is a rigid object, and the accelerometers are rigidly fixed to it.

For interactive use.
High-pass filtering the accelerometer with a bandwidth of 10Hz - you can't filter it much more than that or you lose important 'wobbles', because you need to integrate them to come up with a position - leads to a noise floor of 300uG/sqrt(Hz) *sqrt(10Hz) = 1mG. (RMS (No, not that RMS))

Neglecting roll for the moment.

1mG is an acceleration of 1cm/s^2.

If the accelerometers are spaced 10cm apart, then the radius between each and the center is 5cm, meaning the circumference of the circle is 30cm.
Integrating over 1s, noise is around 3cm/s^2.

After 1s, if you happen to hit an average noise peak in each accelerometer at the opposite point - something that'll happen once every 5-10 seconds or so, (absolute peaks are much worse) what happens to the pointing?

Well - the velocity reads out as 6cm/s^2 wrong, which means that the position is now out by 3cm, or 10 degrees.

Results

What does this mean though?
Because the differential accelerations are so small, and comparable to the noise floor for gentle turning and twisting, there are basically two ways that they can be useful.

When more or less stationary:

Picking up turning about the gravity axis - if you are looking at the phone with it in front of you at chest level, with accelerometers at top and bottom for example, this will let you pick up sharp rotations of the phone around the vertical axis - keeping the angle the screen keeps with the floor constant.

Roll signal-noise drops as the phone long axis approaches vertical, becoming useless well before it is vertical.

Movements have to be sharp and fairly large.

It will probably not be good enough for keeping a map aligned when you turn in place.

It may be adequate for some games, but again, the movements have to be sharp, it will false signal as you decrease the wanted signal.

When in motion:

It may be adequate to improve positioning of the GPS, when sharply going round corners, or doing high-G aerobatics.

It can fill in _short_ - 3-5s gaps in GPS coverage.

In the best case - phone on its back, accelleration vector down, and turning in a vehicle, you may be able to tell sharp turns.

This is only useful if the phone is handheld, or not fixed in position to the vehicle, or if the vehicle does not go where it points.

And this is completely neglecting the temperature sensitivity.

What it can't do.

Inertial navigation over more than several seconds.

An example of an accelerometer that might be used is the one in the Wiimote, as documented in this page .
This page gives the sensitivity of the accelerometer as around 300ug/sqrt(Hz).